3. Derivatives

# Chapter 3 Review Exercises

*True or False*? Justify the answer with a proof or a counterexample.

**1. **Every function has a derivative.

#### Solution

False.

**2. **A continuous function has a continuous derivative.

**3. **A continuous function has a derivative.

#### Solution

False

**4. **If a function is differentiable, it is continuous.

Use the limit definition of the derivative to exactly evaluate the derivative.

**5. **

#### Solution

**6. **

Find the derivatives of the following functions.

**7. **

#### Solution

**8. **

**9. **

#### Solution

**10. **

**11. **

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**12. **

**13. **

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**14. **

Find the following derivatives of various orders.

**15. **First derivative of

#### Solution

**16. **Third derivative of

**17. **Second derivative of

#### Solution

Find the equation of the tangent line to the following equations at the specified point.

**18. ** at

**19. ** at

#### Solution

Draw the derivative for the following graphs.

**20.**

**21.**

#### Solution

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by , where is measured in hours after midnight, and the height is measured in feet.

**22. **Find and graph the derivative. What is the physical meaning?

**23. **Find . What is the physical meaning of this value?

#### Solution

. At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

Hours after Midnight, August 26 | Wind Speed (mph) |
---|---|

1 | 45 |

5 | 75 |

11 | 100 |

29 | 115 |

49 | 145 |

58 | 175 |

73 | 155 |

81 | 125 |

85 | 95 |

107 | 35 |

**24. **Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?

**25. **Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

#### Solution

-7.5. The wind speed is decreasing at a rate of 7.5 mph/hr